Would it be valid to choose a probability distribution for assumptions based on the actual maximum likelihood of various distributions derived from MLE estimates?

For example, suppose I find MLE estimates of a normal distribution $[ \mu , \sigma^2 ] = [1,4]$, which I then substitute (along with $n$) back into the likelihood formula $L_N =(2 \pi \sigma^2 )^{-n/2} exp(- \sum_{i=1}^n [(x_i - \mu)^2/(2 \sigma^2 )] )$ to obtain a number $\hat{L}_N$. Could I then go through the same process to obtain $\hat{L}_B$ for a beta distribution and directly choose $\max \{ \hat{L}_N , \hat{L}_B \}$ in deciding which of the two distributions would be more appropriate in further analysis?

Would it be valid to choose a probability distribution for assumptions based on the actual maximum likelihood of various distributions derived from MLE estimates?

For example, suppose I find MLE estimates of a normal distribution $[ \mu , \sigma^2 ] = [1,4]$, which I then substitute (along with $n$) back into the likelihood formula $L_N =(2 \pi \sigma^2 )^{-n/2} \exp(- \sum_{i=1}^n [(x_i - \mu)^2/(2 \sigma^2 )] )$ to obtain a number $\hat{L}_N$. Could I then go through the same process to obtain $\hat{L}_B$ for a beta distribution and directly choose $\max \{ \hat{L}_N , \hat{L}_B \}$ in deciding which of the two distributions would be more appropriate in further analysis?